Asymptotic Law of Distribution of Primes of Special Form

Let $\mathbb{N}_0$ be the set of natural numbers whose binary expansions have an even number of $1$'s, and let $\mathbb{N}_1=\mathbb{N} \setminus \mathbb{N}_0$. In this paper, we obtain asymptotic formulas for the number of primes $p$ not exceeding $X$ and such that $p\in \mathbb{N}_i$, $p+1\in \mathbb{N}_j$, where $i$ and $j$ take values 0 and 1 independently of each other.